The PalaisâÂÂSmale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular saddle points. The Palais-Smale condition is a condition on the functional that one is trying to extremize.
In finite-dimensional spaces, the PalaisâÂÂSmale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not take unbounded sets into bounded sets. In the calculus of variations, where one is typically interested in infinite-dimensional function spaces, the condition is necessary because some extra notion of compactness beyond simple boundedness is needed. See, for example, the proof of the mountain pass theorem in section 8.5 of Evans.
A continuously Fréchet differentiable functional from a Hilbert space H to the reals satisfies the PalaisâÂÂSmale condition if every sequence such that:
has a convergent subsequence in H.
Let X be a Banach space and be a Gateaux differentiable functional. The functional is said to satisfy the weak PalaisâÂÂSmale condition if for each sequence such that
there exists a critical point of with